Download A Brief Introduction to Musical Acoustics

A Brief Introduction
to
Musical Acoustics
EE217
R.C. Maher
Spring 2010
Harmonic and Inharmonic Sounds
• Musical instruments with simple oscillators
usually produce periodic waveforms
• Periodic waveforms have a fundamental
frequency, f0, and a harmonic spectrum: spectral
energy just at frequencies that are integer
multiples of f0.
• These harmonic components are called
h
harmonics,
i
overtones,
t
or partials
ti l .
• Some musical instruments produce inharmonic
sounds: bells, drums, etc.
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Pitch
• Musical sounds often have a pitch that is
related
l t d tto the
th sound’s
d’ spectral
t l content
t t
• The pitch of a harmonic sound is usually
close to the fundamental frequency of that
sound
• Inharmonic sounds may have a perceived
pitch, but it is not merely the fundamental
of some harmonic series
3
Organization of Western Music
• Two harmonic sounds with different
f d
fundamental
t l ffrequencies
i can llead
d tto
interesting frequency coincidences among
their partials
• When the fundamentals have a low integer
ratio relationship, this is a consonant
interval
4
2
Consonant Intervals
Unison
1/1
100
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
3rd
5/4
125
250
375
500
625
750
875
1000
1125
1250
1375
1500
1625
1750
1875
2000
4th
4/3
133 33
133.33
266.67
400
533.33
666.67
800
933.33
1066.67
1200
1333.33
1466.67
1600
1733.33
1866.67
2000
2133.33
5th
3/2
150
300
450
600
750
900
1050
1200
1350
1500
1650
1800
1950
2100
2250
2400
Octave
2/1
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
2400
2600
2800
3000
3200
5
Musical Scales and Temperament
• European music is based on the notion of
a diatonic pitch scale.
scale The scale specifies
the allowable musical pitches: 8 scale
steps out of 12.
• Problem: if integer frequency ratios are
used (Just intonation), chords only sound
in tune if based on fundamental (tonic)
pitch. Changing musical “key” is not
possible.
6
3
Equal Tempered Scale
• To solve the musical “key” problem,
k b d iinstruments
keyboard
t
t now use equall
tempered tuning.
• Note frequencies are distributed uniformly
in a logarithmic span:
fn = f0  2n/12
• Just vs. equal tempered tuning:
Unison
3rd
4th
5th
Octave
100
125.0000 133.3333 150.0000 200.0000
100
125.9921 133.4840 149.8307 200.0000
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Equal Temperament vs. Just
3rd
4th
5th
octave
1/1
5/4
1.259921
4/3
1.33484
3/2
1.498307
2/1
100
125
125.99
133.33
133.48
150
149.83
200
200
250
251.98
266.67
266.97
300
299.66
400
300
375
377.98
400.00
400.45
450
449.49
600
400
500
503.97
533.33
533.94
600
599.32
800
500
625
629.96
666.67
667.42
750
749.15
1000
600
750
755.95
800.00
800.90
900
898.98
1200
700
875
881.94
933.33
934.39
1050
1048.81
1400
800
1000
1007.94
1066.67
1067.87
1200
1198.65
1600
900
1125
1133.93
1200.00
1201.36
1350
1348.48
1800
1000
1250
1259.92
1333.33
1334.84
1500
1498.31
2000
1100
1375
1385.91
1466.67
1468.32
1650
1648.14
2200
1200
1500
1511.91
1600.00
1601.81
1800
1797.97
2400
1300
1625
1637.90
1733.33
1735.29
1950
1947.80
2600
1400
1750
1763.89
1866.67
1868.78
2100
2097.63
2800
1500
1875
1889.88
2000.00
2002.26
2250
2247.46
3000
1600
2000
2015.87
2133.33
2135.74
2400
2397.29
3200
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4
Rhythm
• Beats per minute
• Beats per measure (time signature)
• Duration of musical notes specified in
fractions:
whole, half, quarter, eighth, sixteenth, 32nd
9
Musical Notation
• Notation specifies pitches, durations, and
ti
time
evolution
l ti
• Representation is like a spectrogram:
frequency vs. time
10
5
Standard Tuning Frequencies
“middle C”
 ≈ 78cm
 ≈ 33cm
 ≈ 66cm
C6 1046.5Hz
C5 523.25Hz
A4 440Hz
 ≈ 1.3m
C4 261.62Hz
C2 65.41Hz
 ≈ 2.6m
C3 130.81Hz
 ≈ 5.2m
11
Musical Timbre
• The relative spectral energy at different
f
frequencies
i is
i perceived
i d as a di
distinct
ti t tone
t
color, or timbre (pronounced as either tam-burr or tim-burr)
• Timbre: The combination of qualities of a
sound that distinguishes it from other
sounds of the same pitch and volume
12
6
Musical Instruments
• Almost any object can be considered a
musical instrument
• Most conventional musical instruments
have
– an excitation source
– a vibrating element
– a resonant body
– a means of coupling the vibrations so that
they radiate into the air as sound waves
13
Musical Instruments (cont.)
• The excitation is a motive force
• The vibrating element usually creates
many harmonics
• The resonant body emphasizes some
frequencies and deemphasizes others
• The coupling means takes energy from the
vibrating element and “loses” it (radiates)
into an acoustical wave through the air
14
7
Example: Singing Voice
Projection from the mouth
Resonance of the throat,
nasal passages,
and the mouth
Glottis
(vocal cords)
Lungs
15
Example: String Instrument
Bow or pluck
excitation.
Vibrating string couples
energy to the hollow
wood body (resonator)
(resonator).
Vibrating body couples
sound into the air
(radiation).
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